Basic Integral Techniques

This section is about the the techniques of integration. Some basic integral techniques will be listed below.

Substitution

Use ${\displaystyle u}$ to substitute some part of ${\displaystyle f(x)}$ and then we have ${\displaystyle \frac{du}{dx}}$, use this to substitute some parts including ${\displaystyle dx}$.

Simple Substitution

or the first method of integration.

"Differential method" is the core of simple substitution. It need us to transform the equation. Simply a few steps to make that:

1. let ${\displaystyle u=g(x)}$
2. find ${\displaystyle u}$ that can make ${\displaystyle du=g^{\prime }(x)dx}$
3. then we get ${\displaystyle \int f[g(x)]g^{\prime }(x)dx=\int f(u)du}$

through this sets of steps, we can make the integrand easy to be integrated.

Complicated Substitution

or the second method of integration.

Integration by Parts

When substitution failed, integration by parts can be considered. Its aim is to differentiate the item that is easier to differentiate, integrate the item that is easier to integrate.

Integration by parts is actually the inverse of the product rule in derivative. It can be derived by steps below:

1. ${\displaystyle (uv{)}^{\prime }=u^{\prime }v+u{v}^{\prime }}$
2. ${\displaystyle uv=\int vdu+\int udv}$
3. ${\displaystyle \int udv=uv-\int vdu}$

When ${\displaystyle \int udv}$ is difficult to find, the formula above could be used.

To choose ${\displaystyle u}$ and ${\displaystyle v}$, a common sequence can be followed:

1. Inverse trigonometric function
2. logarithmic function
3. power function (polynomials)
4. Exponential function
5. Trigonometric function

The first two types of functions are difficult to be integrated, but easy for differentiate. The last two are the opposite.